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1 function [h, hdata] = rbfhess(net, x, t, hdata)
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2 %RBFHESS Evaluate the Hessian matrix for RBF network.
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3 %
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4 % Description
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5 % H = RBFHESS(NET, X, T) takes an RBF network data structure NET, a
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6 % matrix X of input values, and a matrix T of target values and returns
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7 % the full Hessian matrix H corresponding to the second derivatives of
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8 % the negative log posterior distribution, evaluated for the current
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9 % weight and bias values as defined by NET. Currently, the
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10 % implementation only computes the Hessian for the output layer
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11 % weights.
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12 %
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13 % [H, HDATA] = RBFHESS(NET, X, T) returns both the Hessian matrix H and
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14 % the contribution HDATA arising from the data dependent term in the
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15 % Hessian.
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16 %
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17 % H = RBFHESS(NET, X, T, HDATA) takes a network data structure NET, a
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18 % matrix X of input values, and a matrix T of target values, together
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19 % with the contribution HDATA arising from the data dependent term in
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20 % the Hessian, and returns the full Hessian matrix H corresponding to
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21 % the second derivatives of the negative log posterior distribution.
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22 % This version saves computation time if HDATA has already been
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23 % evaluated for the current weight and bias values.
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24 %
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25 % See also
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26 % MLPHESS, HESSCHEK, EVIDENCE
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27 %
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28
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29 % Copyright (c) Ian T Nabney (1996-2001)
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30
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31 % Check arguments for consistency
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32 errstring = consist(net, 'rbf', x, t);
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33 if ~isempty(errstring);
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34 error(errstring);
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35 end
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36
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37 if nargin == 3
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38 % Data term in Hessian needs to be computed
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39 [a, z] = rbffwd(net, x);
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40 hdata = datahess(net, z, t);
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41 end
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42
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43 % Add in effect of regularisation
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44 [h, hdata] = hbayes(net, hdata);
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45
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46 % Sub-function to compute data part of Hessian
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47 function hdata = datahess(net, z, t)
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48
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49 % Only works for output layer Hessian currently
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50 if (isfield(net, 'mask') & ~any(net.mask(...
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51 1:(net.nwts - net.nout*(net.nhidden+1)))))
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52 hdata = zeros(net.nwts);
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53 ndata = size(z, 1);
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54 out_hess = [z ones(ndata, 1)]'*[z ones(ndata, 1)];
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55 for j = 1:net.nout
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56 hdata = rearrange_hess(net, j, out_hess, hdata);
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57 end
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58 else
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59 error('Output layer Hessian only.');
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60 end
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61 return
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62
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63 % Sub-function to rearrange Hessian matrix
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64 function hdata = rearrange_hess(net, j, out_hess, hdata)
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65
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66 % Because all the biases come after all the input weights,
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67 % we have to rearrange the blocks that make up the network Hessian.
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68 % This function assumes that we are on the jth output and that all outputs
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69 % are independent.
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70
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71 % Start of bias weights block
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72 bb_start = net.nwts - net.nout + 1;
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73 % Start of weight block for jth output
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74 ob_start = net.nwts - net.nout*(net.nhidden+1) + (j-1)*net.nhidden...
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75 + 1;
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76 % End of weight block for jth output
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77 ob_end = ob_start + net.nhidden - 1;
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78 % Index of bias weight
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79 b_index = bb_start+(j-1);
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80 % Put input weight block in right place
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81 hdata(ob_start:ob_end, ob_start:ob_end) = out_hess(1:net.nhidden, ...
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82 1:net.nhidden);
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83 % Put second derivative of bias weight in right place
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84 hdata(b_index, b_index) = out_hess(net.nhidden+1, net.nhidden+1);
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85 % Put cross terms (input weight v bias weight) in right place
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86 hdata(b_index, ob_start:ob_end) = out_hess(net.nhidden+1, ...
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87 1:net.nhidden);
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88 hdata(ob_start:ob_end, b_index) = out_hess(1:net.nhidden, ...
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89 net.nhidden+1);
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90
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91 return |
